Exploiting the Robustness on Power-Law Networks Yilin Shen , Nam P. Nguyen, My T. Thai

1. Exploiting the Robustness on Power-Law Networks • YilinShen, Nam P. Nguyen, My T. Thai • Presented by : • YilinShen • Dept. Computer Information Science and Engineering • University of Florida

2. Outline • Motivation: Power-law Networks • Models, Measurement and Threat Taxonomy • Power-Law Random Graph Model • Vulnerability Measurement • Threat Taxonomy • Preliminaries • Uniform Random Failures • Preferential Attacks • Degree-Centrality Attacks

3. Outline • Motivation: Power-law Networks • Models, Measurement and Threat Taxonomy • Power-Law Random Graph Model • Vulnerability Measurement • Threat Taxonomy • Preliminaries • Uniform Random Failures • Preferential Attacks • Degree-Centrality Attacks

4. Motivation: Power-Law Networks Main Property: The number of nodes having k connections is proportional to k-β β is a parameter whose value is typically in the range 1 <β < 4 Many Low Degree Nodes Few High Degree Nodes Internet in December 1998 http://cs.stanford.edu/people/jure/pubs/powergrowth-kdd05.ppt 4

5. More Real Network Examples • Many large-scale real-world networks appear to exhibit a power-law graph • Internet: β = 2.1 • World Wide Web: β = 2.1 • Social Networks: β = 2.3 • Protein-protein interaction networks: β = 2.5

6. Outline • Motivation: Power-law Networks • Models, Measurement and Threat Taxonomy • Power-Law Random Graph Model • Vulnerability Measurement • Threat Taxonomy • Preliminaries • Uniform Random Failures • Preferential Attacks • Degree-Centrality Attacks

7. () Power-law Graph Definition(() Graph G()):

8. Power-Law Random Graph Model • Form a set L containing dv disjoint copy of vertex v (mini-vertices); • Choose a random matching of the elements of L; • For two vertices u and v, there is an edge between them if and only if at least one edge of the random perfect matching was connecting copies of u to copies of v.

9. Outline • Motivation: Power-law Networks • Models, Measurement and Threat Taxonomy • Power-Law Random Graph Model • Vulnerability Measurement • Threat Taxonomy • Preliminaries • Uniform Random Failures • Preferential Attacks • Degree-Centrality Attacks

10. Vulnerability Measurement Total Pairwise Connectivity P (in residual power-law networks after the failures and attacks) Why is Total Pairwise Connectivity an effective measurement? It can control the balance among disconnected components while ensuring the nonexistence of giant components.

11. Outline • Motivation: Power-law Networks • Models, Measurement and Threat Taxonomy • Power-Law Random Graph Model • Vulnerability Measurement • Threat Taxonomy • Preliminaries • Uniform Random Failures • Preferential Attacks • Degree-Centrality Attacks

12. Threat Taxonomy • Uniform Random Failure • Each node in G() fails randomly with the same probability p • Preferential Attack • Each node in G() is attacked with higher probability if it has a larger degree • Degree-Centrality Attack • The adversary only attacks the set of centrality nodes with maximum degrees in G()

13. Outline • Motivation: Power-law Networks • Models, Measurement and Threat Taxonomy • Power-Law Random Graph Model • Vulnerability Measurement • Threat Taxonomy • Preliminaries • Uniform Random Failures • Preferential Attacks • Degree-Centrality Attacks

14. Two Lemmas in Literature M. Molloy and B. Reed (1995)

15. Two Lemmas in Literature (Cont.) F. Chung et al. (2002)

16. Some Fundamental Results • Relations between largest connected component and total pairwise connectivity • Robustness of power-law networks

17. Outline • Motivation: Power-law Networks • Models, Measurement and Threat Taxonomy • Power-Law Random Graph Model • Vulnerability Measurement • Threat Taxonomy • Preliminaries • Uniform Random Failures • Preferential Attacks • Degree-Centrality Attacks

18. Uniform Random Failures

19. The Idea of Proof • Compute the expected degree distribution of graph Gr • Use M. Molloy and B. Reed (1995) to find a threshold β0 • When ββ0, we use the branching process method

20. Visualization • The power-law networks are extremely robusteven when the failure probability is unrealistically large • Even though PLN is affected, the number of node-pairs after failure is close to original PLN • Smaller β is better

21. Outline • Motivation: Power-law Networks • Models, Measurement and Threat Taxonomy • Power-Law Random Graph Model • Vulnerability Measurement • Threat Taxonomy • Preliminaries • Uniform Random Failures • Preferential Attacks • Degree-Centrality Attacks

22. Interactive Preferential Attacks • By choosing a different parameter β′, a node of degree i in G(α,) has probability to be attacked • Main Theorem.

23. Expected Preferential Attacks • To attack the expected c nodes • A node of degree i is attacked with probability • Main Theorem.

24. Visualization • Power-Law Networks will not be affected only when under around expected 13% of nodes are attacked • Smaller βis better

25. Outline • Motivation: Power-law Networks • Models, Measurement and Threat Taxonomy • Power-Law Random Graph Model • Vulnerability Measurement • Threat Taxonomy • Preliminaries • Uniform Random Failures • Preferential Attacks • Degree-Centrality Attacks

26. Degree-Centrality Attacks • The intruders intentionally attack the “hubs”, that is, the set of nodes with highest degrees (larger than x0) • Main Theorem.

27. Visualization • Power-Law Networks will not be affected only when under 5% of degree-centrality nodes are attacked • Smaller βis better

28. Thank you for listening!